Pricing swaps and options on quadratic variation understochastic time change models - discrete observations case ∗

Andrey Itkin †, Peter Carr ‡

October 29, 2009

Abstract

We use a forward characteristic function approach to price variance and volatility swaps andoptions on swaps. The swaps are defined via contingent claims whose payoffs depend on the terminallevel of a discretely monitored version of the quadratic variation of some observable reference process.As such a process we consider a class of Lévy models with stochastic time change. Our analysis revealsa natural small parameter of the problem which allows a general asymptotic method to be developedin order to obtain a closed-form expression for the fair price of the above products. As examples,we consider the CIR clock change, general affine models of activity rates and the 3/2 power clockchange, and give an analytical expression of the swap price. Comparison of the results obtained witha familiar log-contract approach is provided.

∗We thank Arthur Sepp and an anonymous referee for useful comments. We assume full responsibility for any remainingerrors.

†Department of Mathematics, Hill Center for Mathematical Sciences, Rutgers, The State University of New Jersey, 110Frelinghuysen Road Piscataway, NJ 08854-8019, aitkin@math.rutgers.edu

‡Bloomberg LP & NYU, 731 Lexington Avenue, New York, NY 10022, pcarr4@bloomberg.net

1

1 Introduction

The modern theory of asset pricing requires a well-defined notion of stochastic integral in order todescribe the gains from dynamic trading strategies conducted in continuous time. The widest class ofprocesses for which stochastic integrals are defined are semi-martingales, which loosely speaking, ariseas the sum of a bounded variation process and a local martingale. The quadratic variation of a semi-martingale is a continuous time process which loosely speaking, arises by integrating over time thesquared increments of the semi-martingale. The financial markets have recently seen the introductionof contingent claims whose payoffs depend on the terminal level of a discretely monitored version ofthe quadratic variation of some observable reference process. When annualized, this random variableis frequently referred to as the realized variance. The most popular example of such a contract is avariance swap (an example of one is given in Appendix 1). Less liquid examples of such contracts arevolatility swaps and options on realized variance.

The greater liquidity of variance swaps relative to other contracts on realized variance is probablydue to the existence of a well-known strategy for replicating the terminal level of quadratic variationunder idealized conditions. This strategy combines dynamic trading in the underlying with a staticposition in a strip of co-terminal options of all positive strikes. Theoretically, if one can observe theinitial prices of all of these options, one can calculate the theoretical variance swap rate [1]. In reality,market quotes for variance swap rates account for both missing strikes and option market illiquidity.In fact, the illiquidity of deeper out-of-the-money options is usually so pronounced that most marketmakers avoid taking positions in them, thereby adding to the replication error induced by missingstrikes. The resulting possibility of loss is typically then compensated for through the setting of thevariance swap quote.

When replicating variance swaps in this manner, at least two sources of errors can occur in practice:

1. Interpolation/extrapolation error due to the finite number of available option quotes relative tothe continuum of option quotes needed to create the log contract.

2. Errors due to third and higher order powers of daily returns, often due to jumps.

The absence of market option prices suggests the use of parametric models which are capable ofachieving consistency with the observed option prices. Examples of such models include local andstochastic volatility models or combinations of both. The reality of jumps further suggests using moresophisticated jump-diffusion and pure jump models to price swaps and options on quadratic variation.Among multiple papers on the subject, note the following [2, 3, 4, 5]. One can also combine the use ofstochastic volatility models and jump models by subjecting jump processes to stochastic time change.For instance, in [2] variance swaps were priced using a familiar log-contract approach by computinga characteristic function of some jump processes with stochastic time change. The latter could beintroduced either by a known distribution of the stochastic time (as in a celebrated VG model ofMadan and Seneta [6]) or by a given SDE which describes evolution of the stochastic time (as anexample, Carr, Geman, Madan and Yor use as the rate of time change the well-know CIR process [7])

Monte-Carlo methods can be used to price the quadratic variation products within these models.Unfortunately, analytical and semi-analytical (eg. FFT) results are available only for the simplestversions of these models. For instance, Swishchuk [8] uses the change-of-time method for the Hestonmodel to derive explicit formulas for variance and volatility swaps. Also, Carr et. al. [4] proposed amethod of pricing options on quadratic variation in Lévy models via the Laplace transform.

In the present paper we consider a class of models that are known to be able to capture at leastthe average behavior of the implied volatilities of the stock price across moneyness and maturity -time-changed Lévy processes. We derive an analytical expression for the fair value of the quadraticvariation and volatility swap contracts as well as use the approach similar to that of [9] to price options

2

on these products.Our main contribution made in this paper is:

• In contrast to Car and Lee [10] who investigated variance swaps under continuous observationshere we consider variance and volatility swaps under discrete observations.

• We use a forward characteristic function approach and propose a new asymptotic method whichallows an analytical representation for the quadratic variation of a Lévy process with stochastictime change, if the latter is an affine process, and the annualized time between the observationsis relatively small1.

• We consider the activity rate models with a rather general jump specification proposed byCarr and Wu [11]. Using our method we prove (Theorem 1) that under this specification theannualized quadratic variation of the Lévy process with stochastic time determined by a purediffusion process is given by the annualized realized variance times a constant coefficient ξ.This coefficient is determined via derivatives of the characteristic function of the underlyingLévy process.

• We also prove (Theorem 2) that given the above conditions the annualized quadratic variationof the Lévy process under stochastic time determined by a jump-diffusion process is alsogiven by a product of the annualized realized variance and a constant coefficient ξ plus someconstant η which is determined via derivatives of the characteristic function of the underlyingLévy process and jump integrals of the time change process.

• We further extend our results by investigating a more general case when discrete observations ofthe underlying spot price occur over a bigger time interval. We show (Theorem 3) that in thiscase the formulae for the price of the quadratic variation swap acquire two extra terms. Thefirst one p0(τ) is a function of time between observations τ and is determined by a particularmodel of the underlying Lévy process. The last term p2(τ)EQ[V 2] is proportional to the squareof variance and is some kind of convexity adjustment.

• In addition to this general results we derive an analytical representation of the variance andvolatility swaps for some particular models, namely the Lévy processes with the CIR timechange and so-called ”3/2 power” time change. The latter model does not belong to the classof the affine models, therefore this result further extends the proposed approach.

The rest of the paper is organized as follows. In section 2 we define a forward characteristicfunction (FCF) and how the quadratic variation of some stochastic process is related to this function.In section 3 we give a general representation of FCF for the Lévy process with stochastic time change.Section 4 considers in details a particular example of the time change which follows a well-known CIRprocess. First we propose an asymptotic method which allows us to derive an analytical expressionfor the quadratic variation of such a process under an arbitrary Lévy model. As examples, Heston andstochastic skew model of Carr and Wu are considered. Then we are discussing volatility swaps andoptions on the quadratic variation and show how to price them analytically within the framework of theproposed approach. Section 5 generalizes these results for a wide class of the time-change processesthat have an affine activity rate. A general theorem is proved which again provides an analyticalrepresentation of the quadratic variation of such a process. In section 6 we extend our approachto one more class of the stochastic time change processes which follows so-called ”3/2 power” clockchange. We show that despite this model is not affine it still allows variance swaps to be priced in aclosed form. Based on the results obtained a comparison of various models with respect to modelingvariance swaps is provided in section 7. We examine the Heston model (Black-Scholes with the CIRtime change, SSM model and NIG model also with the CIR time change and discuss the results.

1A more precise definition is given in the body of the paper

3

2 Quadratic variation and forward characteristic func-

tion

Let St denote the observable price at time t ≥ 0 of some reference index, which is assumed to bestrictly positive. The discretely monitored quadratic variation of the stochastic process st = log St/S0after N + 1 observations is a random variable defined as follows:

N∑

i=1

(sti − sti−1

)2 (1)

The annualized version of this random variable is defined as:

k

N

N∑

i=1

(sti − sti−1

)2, (2)

ni where k is the number of periods per year eg. 252.Suppose that a variance swap matures at time T and further suppose that the observations are

uniformly distributed over (0, T ) with τ = ti − ti−1 = const,∀i = 1, N . For each dollar of notional,the floating part of the payoff on the variance swap is defined as

EQ

[k

N

N∑

i=1

(sti − sti−1

)2]

. (3)

Like all swaps, the variance swap has zero cost of entry and the magnitude of the fixed payment isdetermined at inception. Assuming no arbitrage, there exists a probability measure Q such that thefixed payment per dollar of notional can be presented as:

QN (s) ≡ EQ[

1T

N∑

i=1

(sti − sti−1)2]

=1T

N∑

i=1

EQ[(sti − sti−1)2

], (4)

Note, that quadratic variation is often used as a measure of realized variance. Moreover, modernvariance and volatility swap contracts in fact are written as a contract on the quadratic variation ofthe log st process because i) this is a quantity that could be observed at the market, and ii) for modelswith no jumps the quadratic variance exactly coincides with the realized variance.

As shown by Hong [12], this fixed payment can be determined in any model where one has knowl-edge of the characteristic function of the future return sti − sti−1 . The idea is as follows.

Let us define a forward characteristic function

φt,T ≡ EQ [exp(iust,T )|s0, ν0] ≡∫ ∞−∞

eiusqt,T (s)ds, (5)

where st,T = sT − st and qt,T is the Q-density of st,T conditional on the initial time state

qt,T (s)ds ≡ Q (st,T ∈ [s, s + ds)|s0) . (6)From Eq. (4) and Eq. (6) we obtain

QN (s) ≡ 1T

N∑

i=1

EQ[(sti − sti−1)2

]=

1T

N∑

i=1

EQ[s2ti,ti−1

](7)

= − 1T

N∑

i=1

∂2φti,ti−1(u)∂u2

∣∣∣u=0

.

4

Thus, if one knows the forward characteristic function of each discrete time increment of the price,one can use the above formula to compute the fixed payment on a variance swap per dollar of notional.The variance swap rate that one quotes is just the square root of this fixed payment.

3 Analytical expression for the forward characteristic

function

According to Carr and Wu [11] consider a d-dimensional real-valued stochastic process Xt|t ≥ 0 withX0 = 0 defined on an underlying probability space (Q,F, P ) endowed with a standard completefiltration F = {Ft|t ≥ 0}. We assume that X is a Lévy process with respect to the filtration F. Thatis, Xt is adapted to Ft, the sample paths of X are right-continuous with left limits, and Xu −Xt isindependent of Ft and distributed as Xu−t for 0 ≤ t ≤ u. The characteristic function of Xt then isgiven by the Lévy-Khintchine theorem (see [13]).

Next, let t → Tt(t ≥ 0) be an increasing right-continuous process with left limits such that foreach fixed t the random variable Tt is a stopping time with respect to F. Suppose furthermore that Ttis finite P -a.s. for all t ≥ 0 and that Tt →∞ as t →∞. Then the family of stopping times Tt definesa random time change. Without loss of generality, we further normalize the random time change sothat E[Tt] = t. With this normalization, the family of stopping times is an unbiased reflection ofcalendar time.

Finally, consider the d-dimensional process Y obtained by evaluating X at T , i.e., Yt ≡ XTt , t ≥ 0.We consider that this process describe the underlying uncertainty of the economy. For example, inthe one-dimensional case, we can take Y as describing the returns on the asset underlying an option.Obviously, by specifying different Lévy characteristics for Xt and different random processes for Ttwe can generate various stochastic processes from this setup. In principle, the random time Tt can bemodeled as a non-decreasing semi-martingale.

In what follows we model dynamics of our underlying spot price by this kind of time-changed Lévyprocess, so that the log return follows the following equation

st ≡ ln St/S0 = (r − q)t + Yt, (8)

where r is the forward interest rate and q is the continuous dividend.To remind, a general Lévy process Xt has its characteristic function represented in the form

φX(u) = EQ[eiuXt

]= e−tΨx(u), (9)

where Ψx(u) is known as a Lévy characteristic exponent ([13]).For time-changed Lévy process, Carr and Wu (2004) show that the generalized Fourier transform

can be converted into the Laplace transform of the time change under a new, complex-valued measure,i.e. the time-changed process Yt = XTt has the characteristic function

φYt(u) = EQ[eiuXTt

]= EM

[e−TtΨx(u)

]= LuTt (Ψx(u)) , (10)

where the expectation and the Laplace transform are computed under a new complex-valued measureM. The measure M is absolutely continuous with respect to the risk-neutral measure Q and is definedby a complex-valued exponential martingale

DT (u) ≡ dMdQ

∣∣∣T

= exp [iuYT + TT Ψx(u)] , (11)

5

where DT is the Radon-Nikodym derivative of the new measure with respect to the risk neutralmeasure up to time horizon T . Moreover, optimal stopping theorem ensures that

Dt(u) = EQ[DT (u)

∣∣∣Ft]

= exp [iuYt + TtΨx(u)] (12)

is a Q martingale and that for all Ft random variable ZT follows

EM[ZT

∣∣∣Ft]

= EQ[MTMt

ZT∣∣∣Ft

]. (13)

Equation (11) reduces the problem of obtaining a generalized Fourier transform of a time-changedLévy process into a simpler problem of deriving the Laplace transform of the stochastic clock. Thesolution to this Laplace transform depends on the specification of the instantaneous activity rate ν(t)and on the characteristic exponents.

Further we again follow the idea of Hong [12]. For the process Eq. (9) we need to obtain theforward characteristic function which is

φt,T (u) ≡ EQ[eiu(log sT−log st)

∣∣∣Ft]

= eiu(r−q)(T−t)EQ[eiu(YT−Yt)

∣∣∣Ft], (14)

where t < T . First, let us consider a single time-change process. The results for a vector version couldbe obtained in a similar way. From the Eq. (13) one has

EQ[eiu(YT−Yt)

∣∣∣Ft]

= EQ[EQ

[eiu(YT−Yt)

∣∣∣Ft]]

(15)

= EQ[EQ

[eiu(YT−Yt)+(TT−Tt)Ψx(u)−(TT−Tt)Ψx(u)

∣∣∣Ft]]

=[EQ

[MTMt

e−(TT−Tt)Ψx(u)∣∣∣Ft

]]= EQ

[EM

[e−(TT−Tt)Ψx(u)

∣∣∣Ft]]

For Markovian arrival rates ν the inner expectation will be a function of ν(t) only.Now let us consider a time-homogeneous time-change processes, for instance, CIR process with

constant coefficients (as it is later specified in the Eq. (33)). With the allowance for the Eq. (10) thelast expression could be rewritten as

EQ[EM

[e−(TT−Tt)Ψx(u)

∣∣∣Ft]]

= EQ[EM

[e−Ψx(u)

R Tt ν(s)ds

∣∣∣νt]]

= EQ[Luθ (Ψx(u))

∣∣∣νt],

where θ =∫ Tt ν(s)ds.

Now for all the arrival rates that are affine, the Laplace transform Luθ (Ψx(u)) is also an exponentialaffine function in νt

Luθ (Ψx(u)) = exp [α(τ, Ψx(u)) + β(τ, Ψx(u))νt] , τ ≡ T − t. (16)and hence

e−iu(r−q)τφt,T (u) = EQ[eiu(YT−Yt)

∣∣∣Ft]

= EQ[exp [α(τ, Ψx(u)) + β(τ, Ψx(u))νt]

∣∣∣νt]

(17)

= eα(τ,Ψx(u))EQ[eβ(τ,Ψx(u))νt

∣∣∣νt]

= eα(τ,Ψx(u))φνt (−iβ(τ, Ψx(u))νt) .

Here as φνt() we denote the generalized characteristic function of the activity rate process underthe risk neutral measure Q. If this characteristic function is available in a closed form as well asthe characteristic exponent of the Lévy process, then one can use the Eq. (7) with t = ti−1, T = ti(so θ =

∫ titi−1 ν(s)ds) and get an analytical expression for the quadratic variation of the Lévy process

QN (s). Actually if the arrival rate is affine then φνt() is also an exponential affine function.

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4 CIR clock change

Let consider two moments of time: t and t + h, h > 0. In the case of the CIR clock change (dyt =κ(θ−yt)dt+η√ytdZt) the conditional Laplace transform (or moment generation function) of the CIRprocess

ψt,h(v) = EQ[e−vyt+h

∣∣∣yt], v ≥ 0 (18)

can be found in a closed form (see, for instance, [14]). Since νt in our case is a positive process, theconditional Laplace transform characterizes the transition between t and t + h ([15]). The CIR is theaffine process, therefore

ψt,h(v) = EQ[e−vyt+h

∣∣∣yt]

= exp [−a(h, v)yt − b(h, v)] , (19)

where functions a, b obey the differential equations

∂a(h, v)∂h

= −κa(h, v)− 12η2a2(h, v) (20)

∂b(h, v)∂h

= κθa(h, v)

with initial conditions a(0, v) = v, b(0, v) = 0.This system of equations has the following solution

a(h, v) =ve−κh

1 + vη2

2κ(1− e−κh)

(21)

b(h, v) =2κθη2

log[1 + v

η2

2κ(1− e−κh)

].

Now we apply these results to the Eq. (17). We substitute τ = ti − ti−1 instead of h in the aboveexpression and from the Eq. (19) obtain

EQ[eβ(τ,Ψx(u))νt

∣∣∣νt]

= exp[− a(τ,−β(τ, Ψx(u)))νt − b(τ,−β(τ, Ψx(u)))

], (22)

where coefficients a, b are given in the Eq. (21). Therefore, from the Eq. (17)

φti−1,ti(u) = exp[α(τ, Ψx(u))− a(τ,−β(τ, Ψx(u)))νt − b(τ,−β(τ, Ψx(u)))

]. (23)

Now, expressions for α(τ, Ψx(u)) and β(τ, Ψx(u)) in the case of the CIR time-change have beenalready found in [11] and read

β(τ, Ψx(u)) = − 2Ψx(u)(1− e−δτ )

(δ + κQ) + (δ − κQ)e−δτ , (24)

α(τ, Ψx(u)) = −κQθ

η2

[2 log

(1− δ − κ

Q

2δ(1− e−δτ )

)+ (δ − κQ)τ

],

where δ2 = (κQ)2 + 2Ψx(u)η2, κQ = κ − iuησρ and σ is a constant volatility rate of the diffusioncomponent of the process2.

2In [11] the authors do not discuss which branch of the complex logarithm function should be used in the above expression

7

Further let us have a more close look at the Eq. (7). Suppose the distance between any twoobservations at time ti−1 and ti is one day. As it is well-known, taking into account an accuratebusiness calendar brings just small corrections to the final quadratic variation value. Therefore,suppose also that these observations occur with no weekends and holidays. Then τi ≡ ti− ti−1 = τ =const. Further we have to use the Eq. (23) with t = ti−1 and T = ti, substitute it into the Eq. (7),take second partial derivative and put u = 0. As this results in a very tedious algebra we use a simpleMathematica program while the resulting expression is still very bulky. To simplify it and make aqualitative analysis of the results transparent below we propose the following asymptotic method toobtain an approximate price of the quadratic variation swap contract.

4.1 Asymptotic method

A detailed analysis of the Eq. (23) shows that the time interval τ enters this equation only as aproduct κτ .

To prove it, note that according to the Eq. (7) we need to compute a second derivative of φti,ti+τat u = 0. If we expand the Eq. (23) into series on u up to the quadratic term, the double coefficientat u2 is just φti,ti+τ (u = 0). One can validate, for instance with Mathematica, that in the obtainedexpression the time interval τ appears only as a product κτ . Intuitively, this could be understoodbecause in the Eq. (23) for φti,ti+τ the time interval τ appears only either as δτ or κτ . And accordingto the definition of δ given after Eq. (24) at u = 0 δ ≡ κ because Ψx(0) = 0. As the expressionfor φti,ti+τ has to contain (δ

′u)

2(u = 0) and δ′′(u)(u = 0), one could expect to see some other termsproportional to τ , like τη2/κ etc.

Now we introduce an important observation that usually κτ ¿ 1. Indeed, according to theresults obtained for the Heston model calibrated to the market data the value of the mean-reversioncoefficient κ lies in the range 0.01− 30. On the other hand, as it was already mentioned, we assumethe distance between any two observations at time ti and ti−1 to be one day, i.e τ = 1/365. Therefore,the assumption κτ ¿ 1 is provided with a high accuracy.

The above means that our problem of computing φ′′u(ti, ti + τ)(u = 0) has two small parameters- u and κτ . And, in principal, we could produce a double series expansion of φ′′u(ti, ti + τ) on boththese parameters. However, to make it more transparent, let us expand the Eq. (23) first into serieson κτ up to the linear terms (that can also be done with Mathematica). Eventually we arrive at thefollowing result

−∂2φti,ti−1(u)

∂u2

∣∣∣u=0

= ξ[θ + (ν0 − θ)e−κti

]τ + O(τ2) (25)

ξ =∂2Ψx(u)

∂u2

∣∣∣u=0

Then from the Eq. (7) we obtain

QN (s) = − 1T

N∑

i=1

∂2φti,ti−1(u)∂u2

∣∣∣u=0

≈ ξT

∫ T0

[θ + (ν0 − θ)e−κt

]dt (26)

= ξ

[θ + (ν0 − θ)1− e

−κT

κT

].

4.2 Some examples

Heston model The above expression could be easily recognized if we remind that the familiar He-ston model can be treated as the pure continuous Lévy component (pure lognormal diffusion process)

8

with σ = 1 under the CIR time-changed clock. For the continuous diffusion process the characteristicexponent is (see, for instance, in [11]) Ψx(u) = σ2u2/2, therefore for the Heston model (Ψx)

′′u(0) = 1

3 and

QN (s) = θ + (ν0 − θ)1− e−κT

κT(27)

Thus, we arrive at the well-known expression of the quadratic variation under the Heston model(see, for instance, [8]).

As it is seen from the Eq. (26) adding jump components to the description of the underlyingstochastic process does not change the ansatz of the dependence QN (s, T ) affecting only the coefficientof the ansatz. This looks to be a new and interesting result. Thus, the dependenceQN (s, T ) is basicallydetermined by the stochastic time-change process, rather than by the Lévy model of the process.

To make it more transparent we can switch some steps in the derivation of the Eq. (27). Indeed,let us now first expand φt,T (u) in the Eq. (17) into series on κτ and (r − q)τ (as the interest rate isusually about 1-10% and τ = 1/365 that the first term is also a small parameter), that yields

φt,T (u) = EQ[1− τ νtΨx(u) + O(τ2)

∣∣∣ν0]. (28)

Then substituting this expression into the Eq. (7) we obtain

QN (s) = τξN∑

i=1

EQ [νti |ν0] ≈ ξEQ[∫ T

0νtdt | ν0

]≡ ξEQ[V ]. (29)

The r.h.s. of this formula differs from the definition of the realized variance just by the constantcoefficient ξ. Therefore, if one uses the CIR stochastic clock - same what is used in the Heston model- the resulting expression for QN (s) will differ from that for the Heston model by the same coefficientξ. That is exactly what we obtained above.

SSM model According to Carr and Wu [11] let us consider a class of models that are known tobe able to capture at least the average behavior of the realized volatilities of the stock price acrossmoneyness and maturity.

We use (Ω,Ft,Q) to denote a complete stochastic basis defined on a risk-neutral probabilitymeasure Q under which the log return obeys a time-changed Lévy process

st ≡ log St/S0 = (r − q)t +(LR

T Rt− ξRTRt

)+

(LL

T Lt− ξLTLt

), (30)

where r, q denote continuously-compounded interest rate and dividend yield, both of which are as-sumed to be deterministic; LR and LL denote two Lévy processes that exhibit right (positive) andleft (negative) skewness respectively; TRt and T

Lt denote two separate stochastic time changes applied

to the Lévy components; ξR and ξL are known functions of the parameters governing these Lévyprocesses, chosen so that the exponentials of LR

T Rt− ξRTRt and LLT Lt − ξ

LTLt are both Q martingales.Each Lévy component can has a diffusion component, and both must have a jump component togenerate the required skewness.

Carr and Wu notice that in principle, this generic specification can capture all of the documentedfeatures, for instance, of the currency options. First, by setting the unconditional weight of thetwo Lévy components equal to each other, we can obtain an unconditionally symmetric distributionwith fat tails for the currency return under the risk-neutral measure. This unconditional propertycaptures the relative symmetric feature of the sample averages of the implied volatility smile. Second,

3The second derivative of the drift term −i(r − q)u on u vanishes.

9

by applying separate time changes to the two components, aggregate return volatility can vary overtime so that the model can generate stochastic volatility. Third, the relative weight of the two Lévycomponents can also vary over time due to the separate time change. When the weight of the right-skewed Lévy component is higher than the weight of the left-skewed Lévy component, the modelgenerates a right-skewed conditional return distribution and hence positive risk reversals. When theopposite is the case, the model generates left-skewed conditional return distribution and negativerisk reversals. Thus, we can generate variations and even sign changes on the risk reversals via theseparate time change. Finally, the model captures the instantaneous correlation between the returnand the risk reversal through the correlations between the Lévy components and the time change.

For model design we make the following decomposition of the two Lévy components in the Eq. (30)

LRt = JRt + σ

RWRt , LLt = J

Lt + σ

LWLt , (31)

where (WRt ,WLt ) denote two independent standard Brownian motions and (J

Rt , J

Lt ) denote two pure

Lévy jump components with right and left skewness in distribution, respectively.We assume a differentiable and therefore continuous time change and let

νRt ≡∂TRt∂t

, νLt ≡∂TLt∂t

, (32)

denote the instantaneous activity rates of the two Lévy components. By definition TRt ,TLt have to

be non-decreasing semi-martingales. We model the two activity rates as a certain affine process. Forinstance, it could be a square-root processes of Heston [16]

dνRt = κR(θR − νRt )dt + ηR

√νRt dZ

Rt , (33)

dνLt = κL(θL − νLt )dt + ηL

√νLt dZ

Lt ,

where in contrast to [11] we don’t assume unconditional symmetry and therefore use different mean-reversion κ, long-run mean θ and volatility of volatility η parameters for left and right activity rates.

We allow the two Brownian motions (WRt ,WLt ) in the return process and the two Brownian

motions (ZRt , ZLt ) in the activity rates to be correlated as follows,

ρRdt = EQ[dWRt dZRt ], ρLdt = EQ[dWLt dZLt ]. (34)

The four Brownian motions are assumed to be independent otherwise.Note that the above definition is pretty wide in sense that it covers a lot of the existing models, in-

cluding Merton jump-diffusion model, Kou double-exponential model, NIG, VG, CGMY, Hyperbolic,LS and even pure continuous models.

Now assuming that the positive and negative jump components are driven by two different CIRstochastic clocks as in the Eq. (33), it could be shown in exactly same way as we did for the singletime process, that the annualized fair strike QN (s, T ) is now given by the expression

QN (s, T ) = ξL[θL + (νL0 − θL)

1− e−κLTκLT

]+ ξR

[θR + (νR0 − θR)

1− e−κRTκRT

]. (35)

So now we have two independent mean-reversion rates and two long-term run coefficients that canbe used to provide a better fit for the long-term volatility level and the short-term volatility skew,similar to how this is done in the multifactor Heston (CIR) model.

10

4.3 Volatility swaps

Similar to a contract on the quadratic variation, a volatility swap contract makes a bet on theannualized realized volatility that is defined as follows

V ol(st) ≡ 1TEQ

√√√√N∑

i=1

[ln

stisti−1

]2≈ 1

TEQ

√∫ T0

νtdt∣∣∣ ν0

= 1

TEQ[

√V ], (36)

where V stays for the total annualized realized variance.Swishchuk [8] uses the second order Taylor expansion for function

√V obtained in [17] to represent

EQ[√

V ] via EQ[V ] and V ar[V ] as

EQ[√

V ] ≈√EQ[V ]− V arV

8(EQ[V ])3/2. (37)

As we already showed in the Eq. (29) for the CIR time-change the quadratic variation process Vdiffers from that of the Heston model by the constant coefficient ξ. Therefore, V ar[V ] in our casediffers from that for the Heston model by the coefficient ξ2. As Swishchuk and Brokchaus and Longshowed for the Heston model

V ar[V ] =η2e−κT

2κ3T 2[(

2e2κT − 4κTeκT − 2) (ν0 − θ) + θ(2κTe2κT − 3e2κT + 4eκT − 1)] . (38)

Thus, for the Lévy models with the CIR time-change the fair value of the annualized realizedvolatility is

V ol(st) =√

ξ V olH(st), (39)

where V olH(st) is this value for the Heston model obtained by using the Eq. (37), 38 and 27.A more rigorous approach is given by Jim Gatheral [18]. He uses the following exact representation

EQ[√

V]

=1

2√

π

∫ ∞0

1− EQ[e−xV

]

x3/2dx. (40)

Here

EQ[e−xV

]= EQ

[exp

{−x

∫ T0

vtdt

}]

is formally identical to the expression for the value of a bond in the CIR model (Eq. (19)) if onesubstitutes there β(τ, Ψx(u)) with −x.

4.4 Options on the quadratic variation

Having known the values of EQ[V ] and EQ[√

V ] we can price vanilla European options on the quadraticvariation using a log-normal method of Jim Gatheral [5]. This method, however, first is an approx-imation, and second, for complicated models like SSM, accurate computing of EQ[

√V ] could be a

problem. Also we found after a careful consideration that another method proposed in [4] results justin a true identity, and thus cannot be used for obtaining the option value. Therefore we intend toproceed in sense of Roger Lee paper [19] and make use of the FFT method.

Let us denote

Q(T ) ≡ λ∫ T

0νtdt, λ ≡ (Ψx)′′u(0)

1T

. (41)

11

For the CIR process the characteristic function φ(u, T ) ≡ EQ[eiuQ(T )] is known

φ(u, T ) = AeB, (42)

B =2iuλv0

κ + δ coth(δT/2),

A = exp[κ2θT

η2

] [cosh(δT/2) +

κ

δsinh(δT/2)

]−2κθη2

δ2 = κ2 − 2iuλη2.

Therefore, according to [19] the call option value on the quadratic variation is given by the followingintegral

C(K, T ) =e−α log(K)

π

∫ ∞0

Re[e−iv log(K)ω(v)

]dv, (43)

where

ω(v) =e−rT φ(v − iα, T )

(α + iu)2(44)

The integral in the first equation can be computed using FFT, and as a result we get call optionprices for a variety of strikes. For complete details see [19] and Carr and Madan original paper [9].

The put option values can just be constructed from the Put/Call symmetry.Parameter α in the Eq. (43) must be positive. Usually α = 0.75 works well for various models. It

is important that the denominator in Eq. (44) has only imaginary roots while integration in Eq. (43)is provided along real v. Thus, the integrand of Eq. (43) is well-behaved.

Note that a similar approach was proposed in [20].

5 Other affine activity rates models

Further we follow Carr and Wu [11] to consider affine activity rate models with more general jumpspecification. First they prove the following proposition.

Proposition 5.1 (Carr-Wu). If the instantaneous activity rate νt, the drift vector µ(Z), the diffusioncovariance matrix σ(Z)σ>(Z), and the arrival rate γ(Z) of the Markov process are all affine in Zthen the Laplace transform LTt(λ) is exponential-affine in z0.

The assumptions of this proposition mean that the following representation takes place

ν(t) = b>v Zt + cv, b ∈ Rk, cv ∈ R (45)µ(Zt) = a− kZt, k ∈ Rkxk, a ∈ Rk[

σ(Zt)σ(Zt)>]ii

= αi + β>i Zt, αi ∈ R, βi ∈ Rk[σ(Zt)σ(Zt)>

]ij

= 0, i 6= jγ(Zt) = aγ + b>γ Zt, αγ ∈ R, βγ ∈ Rk

In a one factor setting Carr and Wu adopt a generalized version of the affine term structure modelproposed by Filipovic [21], which allows a more flexible jump specification. The activity rate process

12

νt is a Feller process with generator

Af(x) = 12σ2xf

′′(x) + (a′ − kx)f ′(x) (46)

+∫

R+0

[f(x + y)− f(x)− f ′(x)(1 ∧ y)] (m(dy) + xµ(dy)),

where a′ = a+∫R+0

(1∧ y)m(dy) for some constant numbers σ, a ∈ R+, k ∈ R+ and nonnegative Borelmeasures m(dy) and µ(dy) satisfying the following condition:

∫

R+0

(1 ∧ y)m(dy) +∫

R+0

(1 ∧ y2)µ(dy) < ∞. (47)

The first line in Eq. (46) is due to the continuous part of the process and is equivalent to the Cox etal. [22] or Heston [16] specification. The second line is due to the jump part of the process. All threecomponents of the Lévy triplet depend linearly on the state variable x. The condition in Eq. (47)says that the jump component dictated by the measure m(dy) has to exhibit finite variation, whilethe jump component dictated by the measure µ(dy) only needs to exhibit finite quadratic variation.Carr and Wu provide various Lévy measure specifications that can be adopted with the only slightmodification: arrival rates of negative jumps need to be set to zero to have the stochastic clock to bea Lévy subordinator.

Under such a specification, the Laplace transform of random time is exponential

LuTt(Ψx(u)) = exp [−α(t, Ψx(u))− β(t,Ψx(u))νt] , (48)with the coefficients α(t,Ψx(u)), β(t,Ψx(u)) given by the following ordinary differential equations:

β′t(t,Ψx(u)) = Ψx(u)− kβ(t, Ψx(u))−12σ2β2(t,Ψx(u)) (49)

+∫

R+0

[1− e−yβ(t,Ψx(u)) − β(t, Ψx(u))(1 ∧ y)

]µ(dy),

α′t(t,Ψx(u)) = aβ(t,Ψx(u)) +∫

R+0

[1− e−yβ(t,Ψx(u))

]m(dy),

with boundary conditions β(0) = α(0) = 0.Now we are ready to formulate a more general result for the quadratic variation of such Lévy

processes.

Theorem 1. Given the above conditions the annualized quadratic variation of the Lévy process understochastic time determined by a pure diffusion process is

QN (s) = 1T

ξEQ[V ], (50)

ξ ≡ (Ψx)′′u(0)∂2β(t,Ψx(u))

∂t∂u

∣∣∣t,u=0

+ (Ψx)′2u (0)

∂3β(t, Ψx(u))∂t∂2u

∣∣∣t,u=0

.

Proof 1.

We prove it based on the idea considered in the previous sections. Namely, we again express QN (s)as in the Eq. (7) via the forward characteristic function φti−1,ti(u), which is (the drift term alreadyappears as k in the Eq. (49))

φti−1,ti(u) = EQ[LuTτ (Ψx(u))

∣∣∣ ν0]

= EQ[exp [−α(τ, Ψx(u))− β(τ, Ψx(u))νt]

∣∣∣ ν0]. (51)

13

Let us remind the reader that κτ ¿ 1 is a small parameter as well as (r − q)τ ¿ 1. Therefore weexpand the above expression in series on τ up to the linear terms to obtain

φti−1,ti(u) = EQ{

exp [−α(0, Ψx(u))− β(0, Ψx(u))νt] · (52)[1−

(∂α(τ, Ψx(u))

∂τ

∣∣∣τ=0

+∂β(τ, Ψx(u))

∂τ

∣∣∣τ=0

νt

)τ]

+ O(τ2).}

Note, that according to the boundary conditions α(0,Ψx(u)) = β(0, Ψx(u)) = 0. Therefore,differentiating the above formula twice on u gives

∂2φti−1,ti(u)∂u2

∣∣∣u=0

= −τ{

Ψ′′x(0)[∂2α(t, Ψx(u))

∂t∂u

∣∣∣t,u=0

+ νt∂2β(t,Ψx(u))

∂t∂u

∣∣∣t,u=0

](53)

+ (Ψ′x(0))2

[∂3α(t, Ψx(u))

∂t∂2u

∣∣∣t,u=0

+ νt∂3β(t,Ψx(u))

∂t∂2u

∣∣∣t,u=0

] }

Further it could be easily checked from the Eq. (49) and the boundary conditions that β′t(0, Ψx(0)) =0, α′t(0, Ψx(0)) = 0. Differentiating the second equation in the Eq. (49) we obtain that

∂2α(t,Ψx(u))∂t∂u

∣∣∣t,u=0

=∂α

∂u(0, Ψx(0))β(0, Ψx(0)) +

∂β

∂u(0, Ψx(0))α(0,Ψx(0)) (54)

+∫

R+0

y∂β

∂u(0, Ψx(0))e−yβ(0,Ψx(0))m(dy) =

∂β

∂u(0, Ψx(0))

∫

R+0

y m(dy)

∂3α(t,Ψx(u))∂t∂2u

∣∣∣t,u=0

=∂2α

∂u2(0, Ψx(0))β(0, Ψx(0)) +

∂2β

∂u2(0,Ψx(0))α(0, Ψx(0))

+ 2∂α

∂u(0, Ψx(0))

∂β

∂u(0,Ψx(0)) +

∫

R+0

m(dy)e−yβ(0,Ψx(0))[y∂2β

∂u2(0, Ψx(0))

− y2(

∂β

∂u(0, Ψx(0))

)2 ]=

∂2β

∂u2(0, Ψx(0))

∫

R+0

y m(dy)−(

∂β

∂u(0, Ψx(0))

)2 ∫

R+0

y2 m(dy)

Therefore, if the activity rate is a pure diffusion process these two derivatives of α(t,Ψx(u)) vanish.Thus, we finally found that

∂2φti−1,ti(u)∂u2

∣∣∣u=0

= −EQ[ξτνt]. (55)

Using this formula together with the Eq. (7) we obtain the result of the Theorem.

QN (s) = − 1T

N∑

i=1

∂2φti−1,ti(u)∂u2

∣∣∣u=0

≈ 1T

ξEQ[∫ T

0νtdt

∣∣∣ ν0] = 1T

ξEQ[V ]. (56)

¥

Theorem 2. Given the above conditions the annualized quadratic variation of the Lévy process under

14

stochastic time determined by a jump diffusion process is

QN (s) = 1T

ξEQ[V ] + η, (57)

η ≡ (Ψx)′′u(0)∂2α(t,Ψx(u))

∂t∂u

∣∣∣t,u=0

+ (Ψx)′2u (0)

∂3α(t,Ψx(u))∂t∂2u

∣∣∣t,u=0

= (Ψx)′′u(0)

∂β

∂u(0, Ψx(0))I1 + (Ψx)′2u (0)

[∂2β

∂u2(0, Ψx(0))I1 −

(∂β

∂u(0,Ψx(0))

)2I2

],

In ≡∫

R+0

yn m(dy).

Proof 2. The proof directly follows from the previous Theorem.

Thus, in the case of jumps in the activity rate even the exponential affinity of the Laplace transformof the random time is not sufficient to provide the same result, i.e. the quadratic variation differsnot just by a constant multiplier ξ from the total realized variance of the process but also by someconstant η.

Now let us consider the case when discrete observations of the underlying spot price occur over abigger time interval, such that κτ might not be a small parameter.

Theorem 3. Given the above conditions the annualized quadratic variation of the Lévy process understochastic time determined by a pure diffusion process is

QN (s) = 1τ

{p0(τ) +

p1(τ)T

EQ[V ] +p2(τ)

TEQ[V 2]

}(58)

Proof 3.

We start with the Eq. (51) and differentiating it twice on u obtain

φti−1,ti(u)∣∣∣u=0

= EQ

{exp [−α(τ, Ψx(0))− β(τ, Ψx(0))νt] · (59)

[−∂

2α(τ, Ψx(u))∂u2

∣∣∣u=0

− ∂2β(τ, Ψx(u))

∂u2

∣∣∣u=0

νt +(

∂α(τ, Ψx(u))∂u

∣∣∣u=0

+∂β(τ, Ψx(u))

∂u

∣∣∣u=0

νt

)2]}

Show that α(τ, Ψx(0)) = β(τ, Ψx(0)) = 0. Expanding α(τ, Ψx(0)) and β(τ, Ψx(0)) in series on τand noticing that Ψx(0) = 0 we have

α(τ, Ψx(0)) = α(0, 0) + τ∂α(τ, 0)

∂τ+

12τ2

∂2α(τ, 0)∂τ2

+ ... (60)

β(τ, Ψx(0)) = β(0, 0) + τ∂α(τ, 0)

∂τ+

12τ2

∂2β(τ, 0)∂τ2

+ ...

Now

1. α(0, 0) = β(0, 0) = 0 according to the boundary conditions to the Eq. (49).

2. From the second equation of Eq. (49) it follows that in case of no jumps α′τ (τ, 0) = aβ(τ, 0) = 0.In turn from the first equation β′τ (τ, 0) = 0.

3. Differentiating the Eq. (49) in time and using a chain rule we arrive at the conclusion that allhigher derivatives of α(τ, 0) and β(τ, 0) in time vanish as well.

15

Thus the Eq. (59) could be rewritten as

−φti−1,ti(u)∣∣∣u=0

= EQ[p0 + p1νt + p2ν2t

], (61)

p0(τ) =∂2α(τ, Ψx(u))

∂u2

∣∣∣u=0

−[∂α(τ, Ψx(u))

∂u

∣∣∣u=0

]2

p1(τ) =∂2β(τ, Ψx(u))

∂u2

∣∣∣u=0

− 2∂α(τ, Ψx(u))∂u

∣∣∣u=0

∂β(τ, Ψx(u))∂u

∣∣∣u=0

p2(τ) = −[∂β(τ, Ψx(u))

∂u

∣∣∣u=0

]2

And therefore

QN (s) = 1T

N∑

i=1

{p0(τ) + p1(τ)EQ[νti ] + p2(τ)EQ[ν

2ti ]

} ≈ 1T

{Np0(τ) +

p1(τ)τEQ[V ] +

p2(τ)τEQ[V 2]

}

=1τ

{p0(τ) +

p1(τ)T

EQ[V ] +p2(τ)

TEQ[V 2]

}(62)

¥Thus if the time distance between market observations is not small the formula for the price of

quadratic variation swap acquires two extra terms. The first one p0(τ) is a function of time betweenobservations τ and is determined by a particular model of the underlying Lévy process. The last termp2(τ)EQ[V 2] is proportional to the square of variance and is kind of convexity adjustment.

Based on this representation we could reconsider our results obtained in the previous sections, forinstance for the CIR time change model. Expanding coefficients p0(τ), p1(τ), p2(τ) into series on τand keeping the first two terms gives

p0(τ) ≈ −12κθ∂2Ψx(u)

∂u2

∣∣∣u=0

τ2 + O(τ3) (63)

p1(τ) ≈ ∂2Ψx(u)∂u2

∣∣∣u=0

τ +

[12κ

∂2Ψx(u)∂u2

∣∣∣u=0

+2η2

κ

(∂Ψx(u)

∂u

∣∣∣u=0

)2]τ2 + O(τ3)

p2(τ) ≈ −(

∂Ψx(u)∂u

∣∣∣u=0

)2τ2 + O(τ3)

This means that in the first approximation on κτ coefficients p0(τ) and p2(τ) vanish. That is whythe price of the quadratic variation swap is proportional to EQ[V ], i.e. the standard log contract.However, p0(τ) and p2(τ) appear in the second order approximation of the price in κτ . For instance,for the Heston model (a pure diffusion underlying process) these coefficients read

p0(τ) = −12κθτ2 + O(τ3), p1(τ) = τ +

12κτ2 + O(τ3) p2(τ) = O(τ3). (64)

In general it can be shown that for the CIR time change model and a pure diffusion underlyingprocess the coefficient p2(τ) = 0.

6 3/2 power clock change

In this section we consider one more class of the stochastic clock change. Despite it is not affine, itstill allows variance swaps to be priced in a closed form.

16

Originally this model has been proposed in a simple form (long term run coefficient is constant) byHeston [23] and Lewis [24] to investigate stochastic volatility. Here we consider a more general casewhen the long-term run could be either a deterministic function of time, or even a stochastic process.

Let the futures price F of the underlying asset be a positive continuous process. By the martingalerepresentation theorem, there exists a process v such that:

dFtFt

=√

vtdZ̃t, t ∈ [0, T ], (65)

where Z̃ is a Q standard Brownian motion. In particular, let us assume the risk-neutral process forinstantaneous variance to be:

dvt = κvt(θt − vt)dt + ²v3/2t dW̃t, t ∈ [0, T ], (66)

where W̃ is a Q standard Brownian motion, whose increments have known constant correlation ρ ∈[−1, 1] with increments in the Q standard Brownian motion Z̃t, i.e.:

dZ̃tdW̃t = ρdt, t ∈ [0, T ]. (67)

The risk-neutral process Eq. (66) for v has its volatility governed by the known positive constant². The 3/2 power specification for the volatility of v is empirically supported. The v process ismean-reverting with speed of mean reversion κvt, where κ is known. The reason that the speed ofmean-reversion is proportional rather than constant is primarily for tractability. In fact, when θt is adeterministic function of time (let us remind that Heston and Lewis explored just the case θ = const),the process Eq. (66) is more tractable than the usual Heston dynamics, since we will show that thereexists a closed form solution for the characteristic function of the log price. In contrast, for the Hestonmodel where the long run mean θt is a deterministic function of time, there is no closed form formulafor the characteristic function of the log price. As a bonus, when v0 > 0 and the process θ is positive,then the process Eq. (66) neither explodes nor hits zero. In contrast, the Heston process can hit zerofor some parameter values,which is unrealistic. Although our primary motivation for proportionalspeed of mean-reversion is tractability, nonlinear drift in the v process is also empirically supported.

In the Eq. (66), the level towards which v reverts is assumed to be an unknown stochastic processθ. We do assume however that θ is conditionally independent of the two Q standard Brownian motionsZ̃ and W̃ , i.e.:

dθtdZ̃t = 0 = dθtdW̃t, t ∈ [0, T ]. (68)We also assume that the evolution coefficients of θ are independent of F and v and of the Brownianmotions Z̃t and W̃t driving them. We may summarize these two assumptions by saying that θ evolvesindependently of F and v. Other than this independence assumption, we assume nothing about thedynamics of θ, not even its initial level. So long as consistency with independence is maintained, theprocess θ can jump, be arbitrarily path dependent, and can depend on other processes.

As a result, we call our pricing theory robust since the risk-neutral dynamics of θ are not fullyspecified. In the Black Scholes model, both θ and v are constant and equal to each other. The usualapproach for introducing stochastic volatility is to specify a particular stochastic process for v andkeep θ constant. Indeed, assuming that the risk-neutral process for instantaneous variance is:

dvt = κvt(θ − vt)dt + ²v3/2t dW̃t, t ∈ [0, T ], (69)

with θ a known constant, Heston[23] and Lewis[24] solve for the characteristic function of XT inclosed form. Once the characteristic function of XT is known in closed form, it is straightforward tonumerically determine option prices.

17

In our model, one can interpret θ as the process that the instantaneous variance would follow ifthe speed of mean reversion were infinite. When κ is finite, the process θ is instead a state variablethat governs the level of v. Without formally realizing it, Carr and Lee [3] consider the case where κ isinfinite. When κ = ∞, the 3/2 specification for the volatility of variance and the correlation ρ betweenthe two Brownian motions becomes irrelevant. Assuming an independent but otherwise unspecifiedstochastic process for v, Carr and Lee show how to replicate the payoff to volatility derivativesby dynamic trading in standard European options and their underlying futures. An unfortunateimplication of their independence assumption is that implied volatility is always a symmetric functionof the difference between log strike and log forward. In other words, their analysis is consistent withthe existence of an implied volatility smile, but not an implied volatility skew. Since implied volatilitytends to display both smile and skew in most markets, they add independent jumps of known size inorder to generate a skew.

To be consistent with both smile and skew without introducing jumps, we employ a finite κ andperimetrically specify how v tends towards the unspecified process θ. As in Carr and Lee, our pricingof volatility derivatives maturing at T will be perfectly consistent with the implied volatility smileof maturity T . In contrast to Carr and Lee, our analysis requires that the parameters κ, ², and ρbe known constants. Our analysis also requires the state variable v0 be known in order to initiallyprice volatility derivatives. For now, we just naively assume that κ, ², ρ, and v0 are somehow known.Future research will focus on the roles of κ, ², ρ, and v0 in calibrating across maturities or determiningappropriate hedge ratios. It will also explore how the parameters might be learned from the timeseries or how these required inputs might be learned from knowledge of prices of related instrumentssuch as variance swaps or barrier options.

6.1 General analysis

Before we discuss how to apply the above formalism of the forward characteristic function to pricingvariance swaps under the described ”3/2-power” clock change, let us first consider another possibleapproach. Let us assume that the Eq. (65) is valid, and let st ≡ ln

(FtF0

)be the log price relative. Let:

φ(u) ≡ EQ[eiusT |FT ], u ∈ R, (70)

be the characteristic function of the log price relative. Using the law of iterated expectations, wehave:

φ(u) = EQ{

EQ[eiusT |FθT ]|FT}

, (71)

where FθT indicates conditioning on the θ path over [0, T ]. As it is shown in [25] the conditionalcharacteristic function EQ[eiusT |FθT ] depends on the particular θ path only through the sufficientstatistic:

I0 ≡∫ T

0eκR t′0 θ(u)dudt′. (72)

In other words, if two θ paths {θ1(u), u ∈ [0, T ]} and {θ2(u), u ∈ [0, T ]} lead to the same value of I0,then the value of the conditional characteristic function is the same. As a result, (71) implies thatthe unconditional characteristic function has the form:

φ(u) =∫ ∞

0ψ(u, I0)q(I0)dI0, (73)

18

for all u ∈ R, where ψ(u, I0) is the conditional characteristic function and q(I0) is the unknownrisk-neutral density of I0. From [25] the conditional characteristic function is given by:

ψ(u, I0) = eiuxΓ(γ̃ − α̃)

Γ(γ̃)

(2

²2Itv

)α̃M

(α̃; γ̃;

−2²2Itv

). (74)

where M(α̃; γ̃; z) denotes the confluent hypergeometric function of the first kind, and:

α̃ ≡ −(

12

+κ̃

²2

)+

√(12

+κ̃

²2

)2+

u(u + i)²2

,

γ̃ ≡ 2[α̃ + 1 +

κ̃

²2

],

and where:κ̃ ≡ κ− ρ²iu.

In (73), we treat the LHS as a known function of u obtained from the market prices of Europeanoptions of all strikes maturing at T . We think of the RHS as an integral transform of the unknownrisk-neutral density q(I0). Since the kernel ψ(u, I0) is known for all u ∈ R and all I0 > 0, one cantheoretically invert for q(I0).

Now consider the problem of determining the Laplace transform of the risk-neutral density of therealized quadratic variation:

L(λ) ≡ EQ[e−λR T0 vtdt|FT ], λ > 0. (75)

Again, using the law of iterated expectations, we have:

L(λ) = EQ{

EQ[e−λR T0 vtdt|FθT ]|FT

}, (76)

where FθT again indicates conditioning on the θ path over [0, T ]. As shown in [25] the conditionalLaplace Transform EQ[e−λ

R T0 vtdt|FθT ] depends on the particular θ path only through the sufficient

statistic I0 defined in (72). As a result, (76) implies that the unconditional Laplace transform has theform:

L(λ) =∫ ∞

0CL(λ, I0)q(I0)dI0, (77)

for all λ > 0, where CL(λ, I0) is the conditional Laplace transform of the risk-neutral PDF of therealized variance and where q(I0) is the now known risk-neutral density of I0. The conditional Laplacetransform of the risk-neutral density of the realized quadratic variation reads ([25])

CL(λ, It) ≡ L(t, v) = Γ(γ − α)Γ(γ)(

2²2Itv

)αM

(α; γ;

−2²2Itv

), (78)

where α and γ are defined after the Eq. (74), and recall that

It ≡∫ T

teκR t′

t θ(u)dudt′. (79)

Hence, by real inversion of this Laplace transform, the risk-neutral density of the realized quadraticvariation can be obtained and hence realized volatility derivatives can be priced.

19

6.2 Closed-form solution for the variance swap

In this section we follow our algorithm that has been described earlier as applied to the affine clockchange in order to derive a closed-form solution for the variance swap price under the stochastic ”3/2-power” clock change. Let us consider Eq. (69), where now θ = θ(t) is a known deterministic functionof time. Again we consider the forward characteristic function of an arbitrary Lévy process with thecharacteristic exponent Ψx(u) under the stochastic clock change determined by the ”3/2-power” law.Similarly to Eq. (51)

φti−1,ti(u) = eiu(r−q)τEQ

[LuTτ (Ψx(u))

∣∣∣ ν0]

= eiu(r−q)τEQ[CL (Ψx(u), Iti)

∣∣∣ ν0], (80)

where

CL(λ, Iti) =Γ(γ − α)

Γ(γ)

(2

²2Itiv

)αM

(α; γ;

−2²2Itiv

), (81)

α = −(

12

+κ

²2

)+

√(12

+κ

²2

)2+ 2

Ψx(u)²2

,

γ ≡ 2[α + 1 +

κ

²2

],

Iti ≡∫ ti

ti−1eκR t′

ti−1 θ(u)dudt′

Now we make an assumption that κθ(t)τ ¿ 1 is a small parameter. This is a generalization of theassumption κτ ¿ 1, that we made for the CIR clock change, for the case of the ”3/2- power” model.Therefore, we expand the above expression in series on κθ(t)τ up to the linear terms.

First of all, expansion of Iti−1 reads

Iti = τ + κθ(τ)τ2 + O(τ2), (82)

and thereforez =

2²2Iti−1vt

≈ 2²2τvt

(83)

As per [26] (13.5.1) an asymptotic expansion series for M(α; γ; z) at large |z| reads

M (α; γ; z) =eiπαΓ(b)Γ(b− a)z

−α[

R−1∑

n=0

(α)n(1 + α− γ)nn!

(−z)−n + O (|z|−R)]

(84)

+ezΓ(b)Γ(a)

zα−γ[

S−1∑

n=0

(γ − α)n(1− α)nn!

(−z)−n + O (|z|−S)]

, −32π < argz <

32π.

We keep the first two terms in these series with n = 0, 1. Further, omitting a tedious algebra andremembering that Γ(0) = ∞ we find that

− ∂2φti−1,ti(u)

∂u2

∣∣∣u=0

= (Ψx)′′u(0)EQ[τvti ]. (85)

Using this formula together with the Eq. (7) we obtain exactly the same result as for the CIRprocess Eq. (29), i.e.

QN (s) = (Ψx)′′u(0)1T

N∑

i=1

EQ[τνi−1 |ν0

] ≈ (Ψx)′′u(0)EQ[

1T

∫ T0

νtdt | ν0]≡ (Ψx)′′u(0)EQ[V ]. (86)

20

The only difference is that now EQ[V ] is computed using the ”3/2-power” law, rather than theCIR process. This can be done by using a representation of the Laplace transform obtained in [25].Indeed, we have

L(t, v) ≡ EQ[e−λ

R Tt vudu

∣∣∣∣vt = v]

, v ≥ 0, t ∈ [0, T ]and thus

EQ[V ] ≡ EQ[∫ T

tvudu

∣∣∣∣ vt = v0]

= −∂L(t, v)∂λ

∣∣∣λ=0

(87)

=[− log

(2

²2IT v0

)+

Γ′(2ν)Γ(2ν)

− 2M (0,1,0)(

0; 2ν;− 2²2IT v0

)−M (1,0,0)

(0; 2ν;− 2

²2IT v0

)]∂α

∂λ

∣∣∣λ=0

=[− log

(2

²2IT v0

)+

Γ′(2ν)Γ(2ν)

−M (1,0,0)(

0; 2ν;− 2²2IT v0

)]2

2κ + ²2,

where

ν = 1 +κ

²2, IT ≡

∫ T0

eκR t′0 θ(u)dudt′,

M (1,0,0)(α, γ, ζ) is the derivative of M(α, γ, ζ) on α, M (0,1,0)(α, γ, ζ) is the derivative of M(α, γ, ζ) onγ, Γ′(2ν) ≡ dΓ(x)/dx|x=2ν , and as follows from [26] (13.1.2) M (0,1,0)(0, γ, ζ) = 0.

As it can be easily validated, at short maturities when T → 0 the integral IT → 0 as well, andfrom Eq. (84)

M (1,0,0)(0, b,−∞) → log(

2²2IT v0

)− Γ

′(2ν)Γ(2ν)

Therefore, EQ[V ] → 0 as expected, i.e. in this limit the equation Eq. (87) is consistent.From a practitioner point of view computing the derivative of the confluent hypergeometric func-

tion on the first parameter could be kind of tricky. One possible way to eliminate this is to make useof the definition of the Kummer function given in [26]. By comparing the series expansion it could beverified that

M (1,0,0) (0; γ;−z) =∞∑

i=1

zi

i(γ)i= −

(z

γ

)2F2[(1, 1); (2, 1 + γ);−z], (88)

where 2F2(a1, ..., ap, b1, ..., bq, z) is the generalized hypergeometric function (HypergeometricPFQ inMathematica notation, or hypergeom in Matlab).

Another approach could be as follows. Let us consider the hypergeometric equation (see, forinstance, [25], Eq.225)

zh′′(z) + (γ − z)h′(z)− αh(z) = 0. (89)According to [25] it has the solution

h(z) =Γ(γ − α)

Γ(γ)(−1)α M(α; γ; z) (90)

Let us differentiate the Eq. (89) on the parameter α, and then put α = 0 to obtain

zw′′(z) + (γ − z)w′(z) = 1, (91)where w(z) = ∂h(z)/∂α, and we took into account that h(z)|α=0 = 1. This equation has the followingsolution

w(z) = C1 + C2I1(z)− I2(z), (92)

I1(z) =∫ z

1

et

tγdt, I2(z) =

∫ z1

et

tγΓ(γ, t)dt

21

where Γ(x, y) is the incomplete gamma function. Differentiating now Eq. (90) on α and comparingwith the Eq. (92) gives

C1 = −Γ′(γ)

Γ(γ)+ M ′α(0; γ; 1), C2 =

I2(0)−M ′α(0; γ; 1)I1(0) (93)

Now from the Eq. (87) we obtain

EQ[V ] =2

2κ + ²2

{− log(z) + Γ

′(2ν)Γ(2ν)

+(

12ν

)2F2[(1, 1); (2, 1 + γ); 1])

[I1(z)I1(0) − 1

]+ I2(z)− I2(0)I1(z)I1(0)

},

(94)

z ≡ 2²2IT v0

As in the Eq. (87) the long-term run coefficient θ(t) is an arbitrary function of time, it gives onea very nice opportunity to better calibrate this model to the real market data.

Another important observation is that, for instance, the CIR model for the stochastic time changeis linear in drift. In other words, the SDE which governs the stochastic variance vt has a drift termlinear in vt. Therefore, let us assume that the instantaneous variance Vt is mean reverting in general,i. e.

dVt = k(θ − Vt)dt + dMt (95)

where dMt is the increment at t of a martingale, e.g. dMt = w(Vt, t)dWt. Then for any choice of M,it is easy to give a closed form expression for the fair strike of the variance swap. Just note that

EQ[dVt] = dEQ[Vt] = k(θ − EQ[Vt])dt (96)

Hence if µ(t) = EQ[Vt], then (96) implies the first order linear ODE µ′(t) = k(θ − µ(t)). Solvingthis subject to µ(0) = V0 and integrating over t from 0 to T gives fair strike of the variance swap.Note that the answer is independent of how the volatility of Vt is specified.

In contrast, when the drift of Vt is nonlinear, e.g. quadratic then the answer depends on how thevolatility of Vt is specified. Our ”3/2-power” process gives one way to proceed.

7 Numerical experiments

As an numerical example first we determine a fair strike of the quadratic variation for three models.

Heston model. Considering the Heston model as a pure diffusion process (GBM with drift µ andvolatility 1) under the CIR time change, the expression for the characteristic exponent of this processreads

Ψx(u) = −iµu + 12σ2u2, (97)

therefore Ψ′′x(u)|u=0 = σ2.The Heston model has 5 free parameters κ, θ, η, ρ, v0 that can be obtained by calibrating the model

to European option prices. In doing so one can use an FFT method as in Carr and Madan [9].

22

SSM. The second model is the stochastic skew model of Carr and Wu that has been briefly describedin section 4. To complete the description of the model we specify two jump components JLt and J

Rt

using the following specification for the Lévy density [11]

µR(x) =

{λRe−|x|/ν

Rj |x|−α−1, x > 0,

0, x < 0.µL(x) =

{0, x > 0,

λLe−|x|/νLj |x|−α−1, x < 0. (98)

so that the right-skewed jump component only allows up jumps and the left-skewed jump componentonly allows down jumps. In contrast to [11] for both type of jumps, we use different parametersλ, νj ∈ R+. This specification has its origin in the CGMY model of Carr, Geman, Madan, and Yor[4]. The Lévy density of the CGMY specification follows an exponentially dampened power law.Depending on the magnitude of the power coefficient α the sample paths of the jump process canexhibit finite activity (α < 0), infinite activity with finite variation (0 < α < 1), or infinite variation(1 < α < 2). Therefore, this parsimonious specification can capture a wide range of jump behaviors.Further we put α = −1, so the jump specification becomes a finite-activity compound Poisson processwith an exponential jump size distribution as in Kou [27].

For such Lévy density the characteristic exponent has the following form

ΨRx (u) = −iuλR[

11− iuνRj

− νRj

1− νRj

]+ ΨRd (u) (99)

ΨLx (u) = iuλL

[1

1 + iuνLj− ν

Lj

1 + νLj

]+ ΨLd (u)

Ψkd(u) =12(σk)2(iu + u2), k = L,R,

where Ψkd(u) is the characteristic exponent for the concavity adjusted diffusion component σWt− 12σ2t.Thus, form Eq. (99) we find that (Ψkx)

′′(0) ≡ (σk)2 + 2λknkj , k = L,R.Overall, the SSM model has 16 free parameters κk, θk, ηk, ρk, vk0 , σ

k, λk, νkj , k = L,R that can beobtained by calibrating the model to European option prices, again using the FFT method.

NIG-CIR. The normal inverse Gaussian distribution is a mixture of normal and inverse Gaussiandistributions. The density of a random variable that follows a NIG distribution X ≈ NIG(α, β, µ, δ)is given by (see [28])

fNIG(x;α, β, µ, δ) =δαeδγ+β(x−µ)

π√

d2 + (x− µ)2K1

(α√

δ2 + (x− µ)2)

, (100)

where K1(w) is the modified Bessel function of the third kind.As a member of the family of generalized hyperbolic distribution, the NIG distribution is infinitely

divisible and thus generates a Lévy process (Lt)t>0. For an increment of length s , the NIG Lévyprocess satisfies

Lt+s − Lt ≈ NIG(α, β, µs, δs) (101)Combined with the CIR clock change it produces a NIG-CIR model. The possible values of the

parameters are α > 0, δ > 0, β < |α|, while µ can be any real number.Below for convenience we use transformed variables, namely:

Θ ≡ β/δ, ν ≡ δ√

α2 − β2

23

The characteristic exponent of the NIG model reads

Ψx(u) = iuµ + δ[√

α2 − β2 −√

α2 − (β + iu)2]

(102)

Calibration. For all the tests given below we first retrieved the vanilla option price data fromhttp://finance.yahoo.com/. If the option was American, we computed its implied volatility andused this value to find a corresponding European vanilla option price. We used just the OTM putsand calls. Then to calibrate the model we minimized a nonlinear least-square functional

minp1...pN

L∑

i=1

wi(Vi,market − Vi,model)2, (103)

where p1...pN are the best fit parameters of the model to be found, Vi,market, i = 1, L is the setof market data, Vi,model, i = 1, L are the corresponding theoretical values, and wi, i = 1, L are theweights. If the characteristic function of the model is known in the closed form we used the Carr-Madan procedure to find the option price via FFT. The weights were defined as by wi = 1/(NS)q,where NS is the normalized strike, NS = log(K/F )/(σATM

√T ), K is the option strike, F is the

forward price, σATM is the option ATM volatility, q is some constant, that in the below tests waschosen q = 2. Therefore, the option prices closer to the ATM acquired a bigger weight. The interestrates were averaged through the time period used in the calibration routine.

The Eq. (103) was solved using differential evolution - a global optimization method which belongsto the class of genetic algorithms. The set of parameters to be calibrated was chosen appropriately.For instance, as it is known, for the Heston model only 4 of the 5 parameters could be independentlyobtained by calibration.

Results. We use these three models to compute the fair value of the quadratic variation contracton SMP500 and Google on August 14, 2006. Parameters of the models were obtained by calibratingthem to the 480 available European option prices. We found the following values of the calibratedparameters (see Tables 1-3)

24

http://finance.yahoo.com/

κ θ η v0 ρ1.572 0.038 0.504 0.019 -0.699

Table 1: Calibrated parameters of the Heston model

κL θL ηL v0L ρL σL λL νL1.2916 0.6515 2.1152 0.3366 -0.9998 0.2077 0.02396 1.8455

κR θR ηR v0R ρR σR λR νR6.7486 1.999 0.0004 0.0002 0.4049 0.0734 0.0029 0.5864

Table 2: Calibrated parameters of the SSM model

κ θ η v0 ρ δ ν Θ µ2.855 0.093 0.787 0.057 -0.987 0.897 7.533 -1.285 0.482

Table 3: Calibrated parameters of the NIGCIR model

It is interesting to see whether the term structure of the variance swap prices computed usingthese models and the values of the parameters obtained by calibration is able to replicate the marketdata. To remind, all the model’s parameter do not depend on time. Therefore, we compared the fairswap price obtained in such a way with that given by the log contract for SPX (Fig. 1) and Google(Fig. 2). The log contract data were obtained from Bloomberg. As it could be seen usage of the SSMmodel slightly improves an agreement with the log contract as compared with the Heston model. But,nevertheless, the difference is substantial, especially at large maturities.

8 Conclusion

In this paper we investigated variance and volatility swaps and options on these instruments underdiscrete observations. We proposed a new asymptotic method which aims to obtain a closed-formexpression for the fair price of these instruments, if the underlying process is modeled by a Lévyprocess with stochastic time change. This is done in two cases.

The first one is when the stochastic time change process belongs to the class of affine processes.We began with the case when the annualized time between the observations is relatively small andconsidered the activity rate models with a rather general jump specification proposed by Carr andWu [11]. Using our method we proved that under this specification the annualized quadratic variationof the Lévy process with stochastic time determined by a pure diffusion process is given by theannualized realized variance times a constant coefficient ξ. This coefficient is determined via deriva-tives of the characteristic function of the underlying Lévy process. The examples given in the paperconsider the CIR clock change for the Black-Scholes model (which is actually the Heston model),NIG model and SSM model. We also proved the Theorem that the annualized quadratic variation ofthe Lévy process under stochastic time determined by a jump-diffusion process is also given by aproduct of the annualized realized variance and a constant coefficient ξ plus some constant η whichis determined via derivatives of the characteristic function of the underlying Lévy process and jump

25

integrals of the time change process. We further managed to extend our results by investigating amore general case when discrete observations of the underlying spot price occur over a bigger timeinterval. We showed in the Theorem 3 that in this case the formulae for the price of the quadraticvariation swap acquire two extra terms. The first one p0(τ) is a function of time between observationsτ and is determined by a particular model of the underlying Lévy process. The last term p2(τ)EQ[V 2]is proportional to the square of variance and is some kind of convexity adjustment. These two extraterms appears only in the second order approximation on the time interval between the observationsτ . Therefore, in the case of rare discrete observations the standard log-contract price (which is, infact, an expectation of the realized variance) is no longer valid. For the particular case of the CIRtime change the second term p2 vanishes even in the second order of approximation in τ .

The second case considered in the paper is when the stochastic time change follows the so-called”3/2 power” process which is not affine. For this model the closed-form expression for the fair priceof the variance and volatility swaps was also obtained in the closed-form.

The above results could be helpful because they allow fast pricing of the above instruments underrather complicated models, which in turn proved to be able to catch many characteristics of theunderlying process. However, given numerical examples and comparison with the market data indicatethat even these complicated models (at least, these particular three models used in our tests) are notable to capture the term-structure of the variance swaps. One possible way of achieving that is aknown approach of considering the long term coefficient of the mean-reverting part of the varianceprocess to be stochastic as well. So for the future it would be interesting to try applying our approachto this kind of models.

26

Appendix: Typical CBOE contract on variance swaps

S&P 500 3-month Variance Contracts CBOE S&P 500 3-month Variance Futures are basedon the realized, or historical, variance of the S&P 500 Index. CBOE S&P 500 3-month VarianceFutures are quoted in terms of variance points, which are defined as realized variance multiplied by10,000. One variance point is worth $50. For example, a variance calculation of 0.06335 would have acorresponding price quotation in variance points of 633.50, and a contract size of $31,675.00 (633.50x $50).

The Final Settlement Value for CBOE S&P 500 3-month Variance Futures is calculated usingcontinuously compounded daily S&P 500 returns over a three-month period, assuming a mean dailyprice return of zero. A ”continuously compounded” daily return (Ri) is calculated from two referencevalues, an initial value Pi and a final value Pi+1, using the following formula:

Ri = ln(

Pi+1Pi

)

Daily returns are accumulated over a three-month period, and then used in a standardized formulato calculate three-month variance. This three-month value is then annualized assuming 252 businessdays per year:

252Ne − 1

Na−1∑

i=1

R2i .

Here Ne is the number of expected S&P 500 values needed to calculate daily returns during thethree-month period. The total number of daily returns expected during the three-month period isNe − 1. Na is the actual number of S&P 500 values used to calculate daily returns during the three-month period. Generally, the actual number of S&P 500 values will equal the expected number ofS&P 500 values. However, if one or more ”market disruption events” occurs during the three-monthperiod, the actual number of S&P 500 values will be less than the expected number of S&P 500 valuesby an amount equal to the number of market disruption events that occurred during the three-monthperiod. The total number of actual daily returns during the three-month period is Na − 1.

27

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29

Figure 1: Fair strike of SPX in Heston, NIGCIR and SSM models. Comparison with a logcontract (as per Bloomberg).

Figure 2: Same for Google

30

IntroductionQuadratic variation and forward characteristic functionAnalytical expression for the forward characteristic functionCIR clock changeAsymptotic methodSome examplesHeston modelSSM model

Volatility swapsOptions on the quadratic variation

Other affine activity rates models3/2 power clock changeGeneral analysisClosed-form solution for the variance swap

Numerical experimentsHeston model.SSM.NIG-CIR.Calibration.Results.

ConclusionAppendix 1S&P 500 3-month Variance Contracts